166 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
Comparing this with (24), it follows that 
Hence, by (38), 
[*, f] = 0 
[a, a] = 0 . 
(C.) In this case, the values of dajdx, dajdy given by (47), (48), are indeterminate. 
For, because [a, a] = 0 , it follows by (38) that [a, f] = 0 , [m, y j = 0 , [a, £] = 0 . 
Hence, to determine dajdx, it is necessary to differentiate (44) with regard to x. 
Therefore 
[&£«] +2 [£«,a]^ + [a, «, a] 
• ( 76 ). 
Hence, because [a, a] = 0 , and assuming that d 2 ajdxd is finite, dajdx satisfies the 
equation 
+[«,«, a] = 0 • • ■ (77) 
\ 0a 
Similarly dajdx satisfies (77). 
Hence, when x = g, y — y, z = £, aq = a 2 = a, it follows that dajdx, dajdx are 
roots of the same quadratic. 
They are finite provided 
WfJUaJ fO .(78). 
The case excluded is that in which Df(x, y, z, a)/Da = 0 is satisfied by three equal 
values of a, when x = y = y, z = £. This case might be investigated in a similar 
manner to the case in which the above equation is satisfied by only two equal values 
of a, when x = y = y, z = £. 
D. Examination of the Differential Coefficients of A. 
In this case A and its differential coefficients are given by equations (51), (52). 
(70), and (71). From these it can be seen, without solving the quadratic (77) for 
da ] /dx, dajdx, that A, dA/dx, d~Ajdx 2 , 0 3 A/0x 3 all vanish, when x = £, y = y, z = £. 
In like manner it can be shown that all the third differential coefficients of A 
vanish ; and, therefore, if U = 0 be the equation of the locus of unodal lines which is 
also an envelope, A contains U 4 as a factor. 
Example 6 .— Locus of Unodal Lines which is also an Envelope. 
Let the surfaces be 
l<f> (z, y, Z )J - bl* ( x > y, z) — «] 3 = 0 . 
